PHYSICS OF FLUIDS
VOLUME 13, NUMBER 12
DECEMBER 2001
Sound–ultrasound interaction in bubbly fluids: Theory and possible
applications
D. B. Khismatullina)
Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington Street,
Boston, Massachusetts 02215
I. Sh. Akhatov
Institute of Mechanics, Ufa (Bashkortostan) Branch of Russian Academy of Sciences, 12 K. Marx Street,
Ufa 450000, Russia
共Received 13 November 2000; accepted 7 September 2001兲
The interaction between sound and ultrasound waves in a weakly compressible viscous liquid with
gas bubbles is considered. Using the method of multiple scales one- and two-dimensional nonlinear
interaction equations are derived. The degeneracy of the interaction is found in bubbly fluids. This
phenomenon lies in the fact that the interaction coefficients vanish at a certain frequency of
ultrasound. We demonstrate that the integrable Davey–Stewartson I 共DSI兲 system of equation can
describe the two-dimensional sound-ultrasound evolution. The DSI equations are remarkable by
their solutions referred to as dromions. In bubbly fluids the dromion represents the localized focused
ultrasound wave which can alter the direction of its motion under changes in the boundary
conditions for the sound wave. The condition of singular focusing of ultrasound in bubbly fluids is
obtained. By numerical analysis of the interaction models, we reveal such processes as
intensification of ultrasound by sound, nonlinear instability of a sound profile, and prove the validity
of the singular focusing condition. Finally, possible applications of the results are outlined. © 2001
American Institute of Physics. 关DOI: 10.1063/1.1416502兴
I. INTRODUCTION
speed of short waves is equal to the phase speed of the long
wave 共the long-wave–short-wave resonance19兲. The resonance requires a special type of dispersion relation, e.g., consisting of two branches, and therefore, occurs in a limited
number of physical systems. Examples of such systems are
waves on the water surface20 and waves in a collision-free
plasma.21 We demonstrate here that the resonance condition
is also satisfied for pressure waves in a bubbly fluid.
The interaction of long and short waves is described by
nonresonant and resonant models. With no damping the nonresonant one-dimensional model consists of the nonlinear
Schrödinger equation for the short-wave envelope and an
algebraic correlation between the long-wave profile and a
square of the envelope modulus.22 The resonant model represents the Zakharov equations.21 The two-dimensional interaction is governed by the Davey–Stewartson 共DS兲 system of
equations.23 There are only two integrable forms of the DS
equations referred to as DSI and DSII equations.24 In the
nonintegrable case the solutions of this system can possess a
blowup instability. As a result, singular focusing takes
place.25,26
The paper is devoted to the theoretical description of the
interaction between sound and ultrasound waves in viscous
bubbly fluids. The nonlinear wave phenomena presented
here, such as degeneracy of the interaction, nonlinear instability of sound waves and singular focusing of ultrasound,
are novel for bubbly fluid dynamics. The remainder of the
paper is structured as follows. In Sec. II the equations of
motion for bubbly fluids and the dispersion relation for plane
sound waves in these media are considered. The method of
multiple scales is described in Sec. III. We demonstrate this
Liquids with gas bubbles have strongly pronounced nonlinear acoustic properties due to nonlinear oscillations of
bubbles and high compressibility of gas. Within recent decades theoretical and experimental investigations have detected many kinds of nonlinear wave phenomena in bubbly
fluids, in particular, the ultrasound self-focusing,1,2 the sound
self-transparency,3 wave front conjugation,4 the acoustic
phase echo,5 subharmonic wave generation,6 the intensification of sound waves in nonuniform bubbly fluids,7,8 the
structure formation in acoustic cavitation,9–11 and differencefrequency sound generation.12,13
To date, long and short waves in bubbly fluids were studied independently. The Korteveg–de Vries 共KdV兲 equation
was proposed14 for describing the evolution of long-wave
disturbances in an ideal liquid with adiabatic gas bubbles.
The existence of the long-wave KdV soliton was confirmed
by many experiments.15,16 The nonlinear Schrödinger 共NLS兲
equation was obtained17 as an equation for a short-wave
modulation in polydisperse bubbly mixtures. The generation
of short-wave subharmonics in bubbly liquids was investigated by a means of geometrical acoustics.6
Despite different length scales, long and short waves can
interact. Benney18 was the first to call attention to the possibility of such a wave interaction on the water surface. The
interaction comes into particular prominence when the group
a兲
Author to whom correspondence should be addressed. Present address:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123.
Electronic mail: damir@math.vt.edu.
1070-6631/2001/13(12)/3582/17/$18.00
3582
© 2001 American Institute of Physics
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Phys. Fluids, Vol. 13, No. 12, December 2001
Sound–ultrasound interaction in bubbly fluids
method on the problem of weakly two-dimensional sound
evolution in a bubbly fluid 共Sec. IV兲. The one-dimensional
interaction equations are proposed in Sec. V. We analyze the
degeneracy of the interaction and propose new equations for
the degenerate resonant interaction in Sec. VI. Section VII
deals with the two-dimensional interaction. We construct the
DS equations and demonstrate that their unique properties as
a dromion solution and singular focusing are true of bubbly
fluids. The numerical modeling of the one-dimensional resonant interaction, two-dimensional defocusing and singular
focusing of ultrasound is conducted in Sec. VIII. We conclude by some words about possible applications of the results in Sec. IX.
w̃⫽
w
,
␷
d␳
⫹ ␳ div v⫽0,
dt
冋
d 2a
冉 冊册
2
3 da
␳l a 2 ⫹
2 dt
dt
␷ ⫽
*
冉 冊
冉 冊
␣ g0 p 0
␳0
t ⫽a 0
*
␳ l ⫺ ␳ l0 ⫽
p⫺ p 0
C 2l
␳ ⫽ ␳ l 共 1⫺ ␣ g 兲 ,
⳵t2
␣ g ⫽ 43 ␲ a 3 n b ,
p
p̃⫽ ⫺1,
p0
˜␳ ⫽
p g /p 0 ⫽ 共 a 0 /a 兲 3 ␬ .
␳⫺␳0
,
␳
*
⫺
u
ũ⫽ ,
␷
*
ỹ⫽
*
y
,
l
*
␳0
p0
共2a兲
1/2
共2b兲
,
1/2
a0
共2c兲
,
共2d兲
,
⳵2p
⳵2p
⳵x
⳵y2
⫺
2
⫽0,
共3a兲
␳ ⫺1⫺b 2 p⫹ 共 1⫹a 兲 3 ⫽0,
共 1⫹a 兲
冉
⳵ 2a
⳵t
2
⫹
冉 冊
3 ⳵a
2 ⳵t
p0
␳ l0 ␣ g0 C 2l
␮ *⫽
Here v⫽ui⫹wj is the velocity vector of the mixture; d/dt
⫽ ⳵ / ⳵ t⫹u ⳵ / ⳵ x⫹w ⳵ / ⳵ y is the substantive derivative with respect to time; p, ␳ are the pressure and mass density of the
mixture; ␳ l is the true density of the liquid; C l is the speed of
sound in the pure liquid; p g , ␣ g , a, and n b are the pressure,
volume content, radius, and number density of the bubbles;
␬ is the polytropic exponent, and the subscript 0 refers to the
unperturbed state of the mixture.
The processes in question suggest the oscillatory regime
of a radial bubble motion. Then all dissipative effects can
take into account on a basis of effective viscosity,29 i.e., a
certain effective coefficient ␮ e f , allowing for the liquid viscosity, ␮ l , thermal damping, ␮ (T) , and other dissipative
mechanisms, is introduced: ␮ e f ⫽ ␮ l ⫹ ␮ (T) ⫹ • • • .
Going over dimensionless variables
a
ã⫽ ⫺1,
a0
⳵ 2␳
共1兲
,
*
x
,
l
and ignoring the terms of order ␣ g0 as compared to unity,
one can reduce the system of Eqs. 共1兲 to the following equations for the perturbations in the pressure p̃, mass density ˜␳
and bubble radius ã:16,30
b⫽
4 ␮ e f da
⫽p g ⫺p,
⫹
a dt
x̃⫽
␳ ⫽ ␳ 0 ␣ g0 ,
*
dn
⫹n b div v⫽0,
dt
dv
␳ ⫹grad p⫽0,
dt
,
where
* 共 ␣ g0 兲 1/2
Consider a mixture of a uniform liquid containing
spherical gas bubbles. Assume that the liquid is weakly compressible, all bubbles have the same radius, the pressure in
the bubbles varies according to the polytropic law. Suppose
that the medium is noncollisional, i.e., we do not take into
account direct interactions and collisions between bubbles
and the processes of bubble breakup, adhesion, and formation. Neglect the effect of external forces as well as the capillary effects. The motion of such a bubbly fluid is described
by the equations27–29
t
*
l ⫽
II. BASIC EQUATIONS
t
t̃ ⫽
3583
冊
2
⫹
共3b兲
␮* ⳵a
⫺ 共 1⫹a 兲 ⫺3 ␬ ⫹p⫹1⫽0,
1⫹a ⳵ t
共3c兲
1/2
4␮ef
a 0 共 p 0 ␳ 0 兲 1/2
共4a兲
,
共4b兲
.
Hereafter the tilde is omitted.
Consider the vector solution to Eqs. 共3兲, z⫽(a,p, ␳ ), in
the form of a longitudinal plane harmonic wave propagating
along x: zÄz0 exp兵i(kx⫺␻t)其, where z0 ⫽(A, P,R) is the constant vector of the solution amplitudes, k and ␻ are the wavenumber and the frequency. The solution is valid when k and
␻ are related by the dispersion relation
冉
␻ 4 ⫹i ␮ * ␻ 3 ⫺ 3 ␬ ⫹
k 2 ⫹3
b2
冊
␻ 2⫺
i ␮ *k 2
b2
␻⫹
3␬k2
b2
⫽0.
共5兲
Later we shall suppose k and ␻ to be non-negative 共the wave
travels from the left to the right兲.
In the nondissipative case, ␮ * ⫽0, relation 共5兲 falls into
two branches 共Fig. 1兲
2
␻⫾
共 k 兲 ⫽ 21 兵 3 ␬ ⫹ 共 k 2 ⫹3 兲 b ⫺2 ⫾ 共关 3 ␬ ⫺ 共 k 2 ⫹3 兲 b ⫺2 兴 2
⫹36␬ b ⫺2 兲 1/2其 .
共6兲
The subscripts ‘‘⫺’’ and ‘‘⫹’’ identify the lower 共or lowfrequency兲 and upper 共or high-frequency兲 branches, respectively.
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3584
Phys. Fluids, Vol. 13, No. 12, December 2001
D. B. Khismatullin and I. Sh. Akhatov
FIG. 1. Dispersion curve for a mixture of water with air bubbles under
normal conditions and with a volume gas content ␣ g0 ⫽1.1⫻10⫺4 .
The long-wave asymptotics of the low-frequency
branch, ␻ l (k l )⫽ ␻ ⫺ 兩 k→0 , and the short-wave asymptotics of
the high-frequency branch, ␻ s (k s )⫽ ␻ ⫹ 兩 k→⬁ , can be written
as follows:
␻ l ⫽c e k l ⫹O 共 k 3l 兲
␻ s ⫽c f k s ⫹O 共 k s⫺1 兲
Let us introduce a parameter ␧Ⰶ1 to satisfy the conditions
p s ⫽␧ s S,
p⫽p l ⫹ p s exp兵 i⌰ 其 ,
L,S⫽O 共 1 兲 .
for k s →⬁,
c e ⫽ ␻ l /k l 兩 k l →0 ⫽ 共 b 2 ⫹ ␬ ⫺1 兲 ⫺1/2,
共7a兲
c f ⫽d ␻ s /dk s 兩 k s →⬁ ⫽b ⫺1
共7b兲
are the equilibrium and frozen speeds of sound in the bubbly
mixture.
Dispersion relation 共6兲 allows the existence of the longwave–short-wave resonance. Actually, the group speed of
short waves
c g ⫽d ␻ s /dk s ⫽k s ␻ s⫺1 共 ␻ s2 ⫺3 ␬ 兲共 2b 2 ␻ s2 ⫺3 ␬ c ⫺2
e
⫺k s2 兲 ⫺1
共8兲
is infinitesimal when k s →0, and it follows from 共2.7兲 that
c f ⬎c e . Therefore, for any sufficiently small k l ⫽k lr , there
exists such a k s ⫽k sr that the long-wave–short-wave resonance condition
共9兲
is satisfied 共the long-dash lines in Fig. 1兲. Here c p (k l )
⫽ ␻ l /k l is the phase speed of the long wave.
The wave numbers of the long and short waves can be
related by the formula k l ⫽␧ k k s , where ␧ k is a certain small
parameter. Upon neglecting the terms of order ␧ k , the resonance condition 共9兲 is simplified
c g 共 k sr 兲 ⫽c e .
III. THE METHOD OF MULTIPLE SCALES
p l ⫽␧ l L,
for k l →0,
where
c g 共 k sr 兲 ⫽c p 共 k lr 兲 ,
Suppose that ␧ k ⫽10⫺3 and consider water with air
bubbles of the radius a 0 ⫽0.1 mm under normal conditions
(p 0 ⫽0.1 MPa, ␳ l0 ⫽103 kg m⫺3 ) and with the volume gas
content ␣ g0 ⫽2.22⫻10⫺4 . The wavelengths of the short
and long waves are then ␭ s ⬇0.67 cm and ␭ l ⬇6.7 m,
respectively. The equilibrium speed of sound C e ⬅c e l /t
* *
⬇730 m s⫺1 . Thus, the frequency of the long wave ⍀ l
⫽C e /␭ l ⬇110 Hz 共audible sound兲. Obviously, the shortwave frequency ⍀ s ⫽ ␻ s /(2 ␲ t )⭓1/(2 ␲ bt )⬇35.6 kHz
*
*
lies in the ultrasound region. Moreover, from 共2c兲 it follows
that a decrease in the bubble radius a 0 leads to increasing the
short-wave frequency ⍀ s . Hence, the long-wave–shortwave interaction in a bubbly fluid can be considered as the
interaction of ultrasound and audible sound propagating in
this medium.
共10兲
It should be noted that the long-wave–short-wave resonance
in bubbly fluids has nothing to do with the resonance of
bubble oscillation which occurs under the condition ␻
⫽ 冑3 ␬ 共for reasonably large bubbles兲. The value 冑3 ␬ corresponds to the horizontal asymptote of the low-frequency
branch. Because the short-wave frequency ␻ s belongs to the
high-frequency branch, it is always above 冑3 ␬ . The Minnaert resonance is, therefore, beyond the scope of this paper.
共11兲
Here p l ,p s are the long- and short-wave pressure perturbations in the mixture; l,s are some numbers 共exponents of
smallness兲; ⌰⫽k s x 0 ⫺ ␻ s t 0 is the phase of the short wave.
We express similarly the perturbations in the bubble radius
and mass density of the mixture. Suppose also that ␧ k ⭐␧.
The nonlinear wave equations describing the interaction
between long and short pressure waves may be obtained using the method of multiple scales.31 According to this
method, the vector solution z to system 共3兲 expands in powers of the parameter ␧ determined above into the long- and
short-wave components
z⫽␧ l
兺
m⭓1
(0)
␧ m⫺1 zm
⫹
兺 兺 ␧ (s⫹m⫺1)n 关 zm(n) e in⌰ ⫹c.c.兴 ,
m,n⭓1
共12兲
and fast (t 0 ,x 0 ,y 0 ) and slow variables (t n ,x n ,y n )
⫽␧ n (t 0 ,x 0 ,y 0 ), where n⫽1,2, . . . , are introduced. The introduction of the slow variables results in the following
asymptotic series:
⳵
⳵
⳵
→
⫹
␧n
,
⳵ ␺ ⳵ ␺ 0 n⭓1 ⳵ ␺ n
兺
␺ ⫽t,x, or y.
共13兲
(0)
(n)
The long- and short-wave components zm
, zm
, m, n
⫽1,2,... depend only on the slow variables. The short wave
is considered to be plane because the phase ⌰ does not contain the fast variable y 0 , but its amplitude may depend on the
slow variables y n , n⫽1,2,... . The numbers (l,s) play a key
role in this asymptotic method because they define a type of
long-wave–short-wave interaction equations.30
We make use of the following ‘‘multiscale-expansion’’
procedure for the derivation of interaction equations:
共1兲 Once we have decided upon the values of l and s, we
substitute the multiscale expansions 共12兲 and 共13兲 into
the equations for perturbations 共3兲. In doing so, we restrict our attention to cubic terms of Eqs. 共3兲, because the
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Phys. Fluids, Vol. 13, No. 12, December 2001
共2兲
共3兲
共4兲
共5兲
higher-order nonlinearities have negligibly small effects
on the long-wave–short-wave interaction.32
The resulting system of equations is split into the zeroth
(n⫽0), first (n⫽1), and second harmonics (n⫽2) and
then, if possible, reduces to one equation for each harmonic of pressure perturbation p.
The algebraic relationship between the second-harmonic
amplitude p (2)
1 and a square of the first-harmonic ampli2
is deduced from the leading order of the
tude 关 p (1)
兴
1
second-harmonic equation.
Using this relationship, we eliminate p (2)
1 from the firstharmonic equation and obtain the coupled system of the
zeroth- and first-harmonic equations. The system is not
(0)
closed yet. We should also eliminate the unknowns p m
(1)
and p m , where m⫽2,3, . . . , from the equations. To do
this would require some assumptions of the unknowns.
The easiest way is to assume that they are equal to zero.
Now, each of the obtained equations contains the terms
of different orders. The next step is therefore splitting the
equations into the orders of ␧. With a few exceptions
共see below兲, we follow this. The expressions appearing
in each order of ␧ are equated to zero. This gives a
bunch of asymptotic equations for the long-wave profile
(1)
p (0)
1 and short-wave envelope p 1 . Among the equations
involving the interaction terms, the lowermost-order
ones represent the equations for long-wave–short-wave
interaction. In particular, the interaction terms of the
lowermost order for the short and long waves are
(1)
2
p (0)
and the second derivative of 兩 p (1)
1 p1
1 兩 共with respect to t or x), respectively. They come from quadratic
nonlinearity. From 共12兲 and 共13兲, it follows that the interaction equations occur in the orders of ␧ l⫹s 共for the
short wave兲 and ␧ 2s⫹2 共for the long wave兲. The lowerorder equations, when present, are linear and without
dispersive terms. 共This can be arranged by choosing the
numbers l and s.兲 They give additional information about
interacting waves: the waves interact in the frame moving with the group speed of short waves and so on. The
issue, which can not be resolved by choosing l and s,
emerges in the case of degeneracy, when the above interaction terms vanish. In the degeneracy case, interaction is of higher order of ␧ than dispersion. We cannot
cut down the dispersive terms and should take into account the interaction terms. The equations for degenerate
interaction therefore contain the terms of different orders
of ␧.
IV. WEAKLY TWO-DIMENSIONAL EVOLUTION OF
SOUND WAVES
Consider the method of multiple scales on the problem
of weakly two-dimensional evolution of sound waves in bubbly fluids. Substitute the multiscale expansions 共12兲 and 共13兲
(n)
⫽0
into Eqs. 共3兲 provided that there is no ultrasound (zm
(0)
and l⫽2, where n⫽1,2, . . . ) and the derivatives of zm with
respect to y are much less than those with respect to x, i.e.,
all unknowns do not depend on y 1 . This means that all
changes in y are slower than in x 共in the direction of wave
Sound–ultrasound interaction in bubbly fluids
3585
motion兲 and can be considered as transverse perturbations in
the wave profile. Then the following equations 共of order ␧ 4 )
关 ␬ 共 1⫺b 2 c 2 兲 ⫺1 兴
⳵ 2 a (0)
1
⫽0,
⳵␰ 2
2 (0)
␳ (0)
1 ⫽⫺3 共 1⫹ ␬ b 兲 a 1 ,
共14兲
(0)
p (0)
1 ⫽⫺3 ␬ a 1 ,
will be true. Here ␰ ⫽x 1 ⫺ct 1 . Equations 共14兲 have a unique
solution when c⫽c e 关cf. 共7a兲兴. Thus, long waves travel with
the equilibrium speed of sound C e ⫽c e l /t in bubbly flu* *
ids. In view of 共14兲 and c⫽c e , the Kadomtsev–Petviashvili
共KP兲 equation33 for the sound profile L⫽ p (0)
1 occurs in the
order of ␧ 6
冉
冊
⳵ ⳵L
⳵L
⳵ 3L
1 ⳵ 2L
⫹2 ␴ L
⫹␹ 3 ⫹
⫽0.
⳵␰ ⳵ ␶
⳵␰
2 ⳵␨ 2
⳵␰
共15兲
Here ␶ ⫽t 2 and ␨ ⫽y 2 . The coefficients ␴ ⫽( ␬
⫹1)c 3e /(4 ␬ 2 ) and ␹ ⫽c 5e /(6 ␬ 2 ) are always positive. When
denoting
T⫽ ␶
冑
␴3
,
27␹
X⫽ ␰
冑
␴
,
3␹
Y ⫽␨
冑
␴3
,
162␹
共16兲
Eq. 共15兲 takes the canonical form
冉
冊
⳵ ⳵L
⳵ L ⳵ 3L
⳵ 2L
⫹6L
⫹ 3 ⫹3 ␴ KP 2 ⫽0
⳵X ⳵T
⳵X ⳵X
⳵Y
共17兲
with ␴ KP⫽1, which is referred to as KPI equation.34 The
sign of ␴ KP may be negative for a number of other physical
systems 共nonlinear optics and water waves25兲 and Eq. 共17兲 is
then known as KPII equation. Both the KPI and KPII are
integrable by means of the inverse scattering transform.34 –36
The simplest soliton solution to 共17兲, regardless of ␴ KP , is
given by
L⫽2V 2 sech2 V 共 X⫺4V 2 T⫺X 0 兲 ,
V,X 0 ⫽const,
共18兲
which represents the famous KdV soliton. This is not surprising, since the KdV equation follows from 共17兲 in the case
that L does not depend on ␨ 共the one-dimensional sound
wave兲:
⳵L
⳵ L ⳵ 3L
⫹6L
⫹
⫽0.
⳵T
⳵X ⳵X3
共19兲
The important question arises of whether such a solitary
wave is unstable in the bubbly mixture with respect to transverse perturbations, i.e., whether these perturbations destroy
the one-dimensional solution 共18兲. There are a number of
analytical25,33,37,38 and numerical works39,40 which showed
that if ␴ KP⫽⫺1 共the KPII case兲, a KdV soliton is linearly
unstable: It transforms into a chain of two-dimensional line
solitons and/or lumps under a periodic transverse perturbation. In contrast, a numerical simulation using the KPI
equation,41 which holds for bubbly mixtures as shown above,
reveals that the solution 共18兲 is stable. Hence, there is no
transverse instability of solitary waves in bubbly liquids. Previously, Gavrilyuk has argued that the KPII equation is valid
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3586
Phys. Fluids, Vol. 13, No. 12, December 2001
D. B. Khismatullin and I. Sh. Akhatov
for bubbly liquids, and therefore, ‘‘solitary waves in bubbly
liquids are linearly unstable.’’42 However, if these waves
were unstable with respect to transverse perturbations, the
experimental results of Nakoryakov and his co-workers14 on
the existence of KdV solitons in bubbly liquids would be
questionable. This is because such experiments would call
for using extremely narrow channels wherein the dependence
of the wave profile on the transverse coordinate could be
neglected. Fortunately, Gavriluk’s conclusion was in error.
V. ONE-DIMENSIONAL INTERACTION
Let us put (l,s)⫽(2,1) and substitute expansions 共12兲
and 共13兲 into Eqs. 共3兲. Suppose that all unknowns do not
depend on y 0 , y 1 , y 2 , . . . 共the one-dimensional case兲. We
then split the resulting expressions into harmonics of the
short wave, exp兵in⌰其, n⫽0,1,2, . . . and restrict our attention
to order ␧ 4 for the zeroth harmonic and ␧ 3 for the first harmonic. We also assume that the damping coefficient ␮ * is of
order ␧ 2 共weak damping兲: ␮ * ⫽␧ 2 ␮ , ␮ ⬃O(1). Then, the
following equations for interaction will be true:
L⫽L 0 ⫹
i
␣
⳵ 2兩 S 兩 2
c 2g ⫺c 2e ⳵␰ 2
共20a兲
,
⳵S
⳵ 2S
⫹ ␤ 2 ⫹i⌫S⫹ ␥ 兩 S 兩 2 S⫽ ␦ LS.
⳵␶
⳵␰
共20b兲
(1)
Here L⫽p (0)
1 is the sound profile; S⫽ p 1 is the ultrasound
envelope. In doing so, L⫽L( ␰ )⫹L 0 , S⫽S( ␶ , ␰ ), ␶ ⫽t 2 , ␰
⫽x 1 ⫺c g t 1 (L 0 is the initial sound profile which is assumed
to be constant兲. The coefficients
␣ ⫽⫺
␤⫽
c 2e c 2g
␬ 共 ␻ s2 ⫺3 ␬ 兲 2
冋
关 ␻ s2 ⫺9 ␬ 共 ␬ ⫹1 兲兴 ,
cg
cg
1⫹ 2
兵 4k s ␻ s ⫹c g 共 k s2 ⫹3 共 1⫹ ␬ b 2 兲
2k s
␻ s ⫺3 ␬
册
⫺6b 2 ␻ s2 兲 其 ,
␦ ⫽⫺
⌫⫽
共21a兲
c g ␻ s2
2 ␬ k s 共 ␻ s2 ⫺3 ␬ 兲 2
␮ ␻ s c g 共 b 2 ␻ s2 ⫺k s2 兲
2k s 共 ␻ s2 ⫺3 ␬ 兲
␥ ⫽⫺
共21b兲
关 ␻ s2 ⫺9 ␬ 共 ␬ ⫹1 兲兴 ,
共 ␻ s2 ⫺3 ␬ 兲 2
共21d兲
,
关 ␻ s2 ⫺3 ␬ 共 3 ␬ ⫹1 兲兴 ␦
共21c兲
⫺
c g 共 b 2 ␻ s2 ⫺k s2 兲
4k s ␻ s2 共 ␻ s2 ⫺3 ␬ 兲 3
⫹27␬ 2 共 ␬ ⫹1 兲 ␻ s2 ⫹27␬ 2 共 ␬ ⫹1 兲 2 兴 .
关 ⫺ ␻ s4
共21e兲
c g and c e are determined from 共8兲 and 共7a兲.
Due to the absence of derivatives in 共20a兲 the resulting
model describes the formation of an inertialess sound by an
ultrasound wave packet and their subsequent interaction.
Substitution 共20a兲 into 共20b兲 and notation S̃⫽S exp兵i␦L0␶其
give the Ginzburg–Landau 共GL兲 equation in S̃
i
⳵ S̃
⳵ 2 S̃
⫹ ␤ 2 ⫹ ␥ ⬘ 兩 S̃ 兩 2 S̃⫹i⌫S̃⫽0,
⳵␶
⳵␰
␥ ⬘⫽ ␥ ⫺
␣␦
c 2g ⫺c 2e
.
共22兲
Generally, the GL equation does not have analytical
solutions.32 But in the nondissipative case (⌫⫽0), Eq. 共22兲
known as the nonlinear Schrödinger 共NLS兲 equation is integrable. The NLS equation possesses soliton solutions43 if the
Benjamin–Feir instability criterion44
␤␥ ⬘ ⬎0,
共23兲
is satisfied. It is worth noting that the Benjamin–Feir instability is the modulational 共temporal兲 instability of spatially
uniform solutions to the nonlinear Schrödinger equation. The
unstable spatially uniform solution is deformed into a train of
localized waves called bright NLS solitons.38 For bubbly fluids, the Benjamin–Feir instability regions were given on the
parameter plane (k s ,b) earlier.16 The physical significance of
this instability in the bubbly fluids may be the following. Let
us assume that a monochromatic ultrasound wave propagates
through the bubbly medium. Its amplitude S does not depend
on x and, therefore, represents the spatially uniform solution
to the NLS equation. Suppose that the condition 共23兲 is satisfied for the wave, i.e., the values for the wave frequency
and bubbly fluid parameters are in the Benjamin–Feir instability region. This monochromatic wave will be unstable to
spatial periodic perturbations. As a result, a train of localized
short-wave packets will be generated with the passage of
time. Hence, it is impossible to sustain a monochromatic
ultrasonic signal in the bubbly fluid if the Benjamin–Feir
instability criterion is satisfied 共spatial pressure perturbations
inevitably occur in experiments兲.
The interaction model 共20兲 becomes incorrect if c g ⫽c e
共the resonant case兲. This is because the term on the righthand side of 共20a兲 goes to infinity. It is necessary to change
the exponents of smallness l and s. Under resonance condition 共10兲 and (l,s)⫽共2, 3/2兲, substitution of expansions 共12兲
and 共13兲 into system 共3兲 results in the Zakharov equations
with damping 共which arises in the orders of ␧ 5 for the zeroth
harmonic and ␧ 7/2 for the first harmonic兲:
⳵L
␣ ⳵兩S兩2
⫹
⫽0,
⳵ ␶ 2c e ⳵␰
i
⳵S
⳵ 2S
⫹ ␤ 2 ⫹i⌫S⫽ ␦ LS.
⳵␶
⳵␰
共24兲
Now L⫽L( ␶ , ␰ ), S⫽S( ␶ , ␰ ), ␶ ⫽t 2 , ␰ ⫽x 1 ⫺c g t 1 .
In contrast to the previous 共nonresonant兲 model, the effects of sound on ultrasound are nontrivial. The dynamics of
resonant sound–ultrasound interaction depends on the initial
conditions for the sound wave.
The Zakharov equations, at ⌫⫽0, is one of the integrable nonlinear wave models. Their spatially uniform solutions are always unstable. The instability evolves for the perturbations with the wave number K⬍(6 冑3 兩 ␣ ␦ 兩兩 S 0 兩 2 ) 1/3. As
a result, there exist the envelope soliton solutions45 for any
values of the coefficients.
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Phys. Fluids, Vol. 13, No. 12, December 2001
Sound–ultrasound interaction in bubbly fluids
3587
From 共27a兲 it follows the invariance of an initial sound profile with respect to the time ␶ . Equation 共27b兲 describes the
linear dispersion and dissipation of the ultrasound wave
packet.
To construct the model of the degenerate resonant interaction we substitute 共12兲 and 共13兲 into 共3兲 and consider all
terms up to order ␧ 6 for the long-wave component and ␧ 9/2
(1)
for the short-wave one. Again L⫽ p (0)
1 and S⫽p 1 are functions of ␰ ⫽x 1 ⫺c g t 1 , ␶ ⫽t 2 . Denote other components as
(0)
(1)
(1)
L 1 ⫽ p (0)
2 , L 2 ⫽ p 3 , S 1 ⫽ p 2 , S 2 ⫽ p 3 and assume that L 1
and S 1 evolve in accordance with Eqs. 共27兲 and L 2
⫽L 2 ( ␰ ), S 2 ⫽S 2 ( ␰ ). Then, under conditions 共25兲 and 共10兲,
the degenerate sound–ultrasound interaction is governed by
the equations
FIG. 2. Coefficients ␣ and ␦ versus the wave number k s ( ␬ ⫽1.4, b
⫽0.447). The white point corresponds to the degeneracy of the interaction.
VI. DEGENERACY OF THE INTERACTION
The NLS and Zakharov equations are classical models
for the long-wave–short-wave interaction. Are there any special features of the sound-ultrasound interaction in bubbly
fluids? It turns out there exists one peculiarity.
In the bubbly fluids, the interaction coefficients ␣ and ␦
can vanish simultaneously 共Fig. 2兲. This happens when the
ultrasound frequency
␻ s ⫽3 关 ␬ 共 ␬ ⫹1 兲兴 1/2.
共25兲
The interaction between sound and ultrasound is then degenerated because the equations for interaction are separated.
However, such a degeneracy does not mean the absence
of interaction. Certainly, the sound profile L( ␰ , ␨ ) is equal to
zero in the nonresonant degenerate case. But if we take into
account the zeroth-harmonic terms up to order ␧ 5 , the inter(0)
action between S⫽p (1)
1 and L 1 ⫽p 2 can occur
L 1⫽
i␭
c 2g ⫺c 2e
冋
S
册
⳵S*
⳵S
⫺S *
,
⳵␰
⳵␰
共26兲
⳵S
⳵ S
i ⫹ ␤ 2 ⫹i⌫S⫹ ␥ 兩 S 兩 2 S⫽0.
⳵␶
⳵␰
2
The coefficient ␭⫽( ␬ ⫹1) 1/2c 2e c 3g / 关 3 ␬ 5/2(3 ␬ ⫹2) 2 兴 never
vanishes. Here the quasimonochromatic ultrasonic signal
will also generate sound but of much smaller intensity than
in the nondegenerate case.
A. Resonant degeneracy: The case „ l , s …Ä„2,3Õ2…
Under condition 共25兲 Eqs. 共24兲 turn into the uncoupled
equations:
⳵L
⫽0,
⳵␶
⳵S
⳵ 2S
i ⫹ ␤ 2 ⫹i⌫S⫽0.
⳵␶
⳵␰
共27a兲
共27b兲
再
冋
⳵L
⳵ 2S
⳵ 2S *
⳵ 3L
⳵L2
⫹␧ ␹ 3 ⫹ ␴
⫺i␭ S * 2 ⫺S
⳵␶
⳵␰
⳵␰
⳵␰
⳵␰ 2
再
册冎
⫽0,
共28a兲
⳵S
⳵ 2S
⳵ 3S
⳵S
⳵S
⫺i ␤ 2 ⫹⌫S⫹␧ ␩ 3 ⫹i ␥ S 2 S * ⫹⌫ ⬘ ⫺ ␯ L
⳵␶
⳵␰
⳵␰
⳵␰
⳵␰
冎
␯ ⳵L
⫺ S
⫽0.
2 ⳵␰
共28b兲
Equations 共28兲 describe the degenerate interaction between
low-intensity sound and ultrasound. This is because the amplitude of the sound wave is less than the ultrasound envelope (l⬎s). Obviously, if S⫽0, Eq. 共28b兲 reduces to the
KdV equation 共19兲 in a more slower time t 3 ⫽␧t 2 .
It should be noted that 共28兲 is determined only by the
parameter ␬ . First, this follows from the degeneracy condition 共25兲 according to which the ultrasound frequency ␻ sr
depends only on ␬ . Second, when 共25兲 is substituted into the
dispersion relation 共6兲, a relation is obtained which relates
the quantities k sr , b, and ␬ . Together with the resonance
condition 共10兲 the latter expression constitutes a set of two
algebraic equations in k sr and b with the parameter ␬ . Hence
k sr , b are completely expressed via ␬
␻ sr ⫽3 关 ␬ 共 ␬ ⫹1 兲兴 1/2,
k sr ⫽
3 共 ␬ ⫹1 兲
␬ 共 3 ␬ ⫹2 兲
1/2
,
共29兲
关 3 ␬ 共 ␬ ⫹1 兲 ⫹1 兴 1/2
.
b⫽
␬ 共 3 ␬ ⫹2 兲
For example, if one considers a mixture of water with air
bubbles of the radius a 0 ⫽0.1 mm, under normal conditions
and the volume gas content ␣ g0 ⬇3•10⫺4 , the degeneracy of
resonant interaction occurs when the ultrasound frequency
f sr ⬇87.5 kHz.
All coefficients of system 共28兲 are positive at ␬ ⭓1
c g ⫽c e ⫽
␴⫽
␬ 共 3 ␬ ⫹2 兲
共 3 ␬ ⫹1 兲共 ␬ ⫹1 兲 1/2
␬ 共 3 ␬ ⫹2 兲 3
4 共 3 ␬ ⫹1 兲 3 共 ␬ ⫹1 兲 1/2
,
共30a兲
,
Downloaded 15 Jun 2002 to 128.113.18.227. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
3588
␭⫽
Phys. Fluids, Vol. 13, No. 12, December 2001
␬ 3/2
6 共 3 ␬ ⫹1 兲 4 共 3 ␬ ⫹2 兲共 ␬ ⫹1 兲
关 27␬ 3 ⫹18␬ 2 ⫹18␬
3/2
⫹4 兴 ,
␤⫽
␩⫽
共30b兲
3 ␬ 5/2共 3 ␬ ⫹2 兲
2 共 ␬ ⫹1 兲 3/2共 3 ␬ ⫹1 兲 3
␬ 3 共 3 ␬ ⫹2 兲
6 共 3 ␬ ⫹1 兲 共 ␬ ⫹1 兲
5
5/2
关 3 ␬ 2 ⫹3 ␬ ⫹2 兴 ,
␹⫽
␯⫽
␬ 3 共 3 ␬ ⫹2 兲 5
6 共 ␬ ⫹1 兲 共 3 ␬ ⫹1 兲
5/2
␬ 共 3 ␬ ⫹2 兲
共 3 ␬ ⫹1 兲 2 共 ␬ ⫹1 兲 1/2
˜␦ ⫽
共30e兲
,
5
3 ␬␮
⌫⫽
,
2 共 3 ␬ ⫹1 兲
,
2 共 ␬ ⫹1 兲 共 3 ␬ ⫹1 兲 共 3 ␬ ⫹2 兲
2
3
共30f兲
2␮
3 ␬ 共 3 ␬ ⫹1 兲共 3 ␬ ⫹2 兲 2
共30g兲
Due to the presence of the damping term ⌫S of lower order
␧ in comparison with the interaction terms, the fast decay of
the ultrasonic signal takes place. This leads to the negligible
action of ultrasound on sound. In order to ‘‘catch’’ effects of
sound on ultrasound, we need large sound amplitudes. But
this seems to contradict the perturbation theory according to
which the sound profile L should be less than ␧ ⫺1 . Consequently, we may argue that weak damping suppresses the
resonant interaction between low-intensity sound and ultrasound in the case of degeneracy.
B. Resonant degeneracy: The case „ l , s …Ä„1, 1…
It turns out that the resonant degenerate interaction can
be described by other models. One of such models occurs
when (l,s)⫽共1,1兲, i.e., in the context of high-intensity sound
in the ultrasonic field.
With choosing (l,s)⫽共1,1兲, the interaction is usually
trivial
L⫽L 0 共 ␰ 兲 ,
i
⳵S
⫽ ␦ LS.
⳵t1
共31兲
But provided that the conditions of resonant degeneracy 共29兲
are satisfied, i.e., ␦ ⫽0, and damping is weak, the following
equations 共of order ␧ 3 for L and S) will govern the interaction in the time ␶ ⫽t 2 :
再
⳵L
⳵L
⳵L兩S兩
⳵L
⳵ L
˜ 兩S兩2
⫹␴
⫹␧ ␹ 3 ⫹␵
⫹%
⫹⌫
⳵␶
⳵␰
⳵␰
⳵␰
⳵␰
3
3
⳵ 2S
⳵ 2S *
⳵␰
⳵␰ 2
⫺S
2
冊冎
⫽0,
%⫽
,
共 3 ␬ ⫹2 兲 2
36␬ 1/2共 ␬ ⫹1 兲 1/2共 3 ␬ ⫹1 兲
共 3 ␬ ⫹2 兲 3
36共 ␬ ⫹1 兲 3/2共 3 ␬ ⫹1 兲 5
共33a兲
,
关 27␬ 4 ⫹62␬ 3 ⫹45␬ 2 ⫹7 ␬ ⫺2 兴 ,
共33b兲
1
6 共 ␬ ⫹1 兲 共 3 ␬ ⫹1 兲 5 共 3 ␬ ⫹2 兲
1/2
关 27␬ 3 ⫹36␬ 2 ⫺4 兴 ;
2
共32a兲
共33c兲
the other coefficients were defined above.
We include the terms of order ␧ in Eq. 共32a兲. Otherwise,
the sound wave will be governed by the Hopf equation
⳵L
⳵L2
⫹␴
⫽0.
⳵␶
⳵␰
关 189␬ 4 ⫹405␬ 3
⫹288␬ 2 ⫹79␬ ⫹4 兴 .
冉
˜⌫ ⫽
关 9 ␬ ⫺2 兴 ,
␬ 3/2␮
⫺i␭ S *
⳵S
⳵ 2S
⳵S
␯ ⳵L
⫹ ␤ 2 ⫺ ␥ S 2 S * ⫹i⌫S⫽i ␯ L ⫹i S
⫹˜␦ L 2 S.
⳵␶
⳵␰
2 ⳵␰
⳵␰
共32b兲
Here
␵⫽
12␬ 1/2共 3 ␬ ⫹1 兲共 3 ␬ ⫹2 兲 2
2
i
共30d兲
共 ␬ ⫹1 兲 1/2
⌫ ⬘⫽
共30c兲
关 81␬ 4 ⫹54␬ 3 ⫹81␬ 2 ⫹60␬
⫹28兴 ,
␥⫽
D. B. Khismatullin and I. Sh. Akhatov
共34兲
The Hopf equation describes the process of shock wave formation from an initially smooth profile. Mathematically this
implies that the derivatives of L and S with respect to ␰ must
go to infinity at some spatial point ␰ ⫽ ␰ in a finite time.
*
Hence, the terms of order ␧, containing these derivatives,
become rather large to be able to influence the dynamics of
sound–ultrasound interaction.
VII. TWO-DIMENSIONAL INTERACTION
Let us consider the two-dimensional interaction, i.e., let
us take into account the dependence of wave perturbations
on the transverse coordinate y. When (l,s) equals (2,1), substitution of multiscale expansions 共12兲 and 共13兲 into Eqs. 共3兲
results in the Davey–Stewartson 共DS兲 equations with damping
共 c 2g ⫺c 2e 兲
i
⳵ 2L
⳵ 2L
⳵ 2兩 S 兩 2
⳵␰
⳵␨
⳵␰ 2
⫺c 2e
2
⫽␣
2
,
⳵S
⳵ 2S
⳵ 2S
⫹ ␤ 2 ⫹% 2 ⫹i⌫S⫹ ␥ 兩 S 兩 2 S⫽ ␦ LS.
⳵␶
⳵␰
⳵␨
共35a兲
共35b兲
(1)
Here L⬅ p (0)
1 ⫽L( ␰ , ␨ )⫹L 0 ; S⬅p 1 ⫽S( ␶ , ␰ , ␨ ); ␶ ⫽t 2 ; ␰
⫽x 1 ⫺c g t 1 ; ␨ ⫽y 1 . The coefficient
%⫽
cg
.
2k s
共36兲
Unlike the one-dimensional case, the resonance condition
c g ⫽c e does not require to change the exponents of smallness
l and s, and the form of equations is conserved.
It should be noted Eq. 共35a兲 does not contain the term
⳵ 2 兩 S 兩 2 / ⳵␨ 2 . This term disappears in going from the radius
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Phys. Fluids, Vol. 13, No. 12, December 2001
Sound–ultrasound interaction in bubbly fluids
3589
(1)
(0)
perturbations a (0)
1 and a 1 to the pressure perturbations p 1
(1)
and p 1 . In particular, the zeroth-harmonic equation for a
has the form
冋
⳵2
⫺c 2e
2
⳵t1
冋
冉
⳵2
⳵2
⳵x1
⳵ y 21
⫹
2
⳵2
⫽ ␣1
⳵ t 21
⫹␣2
冉
冊册
a (0)
1
⳵2
⳵ x 21
⫹
⳵2
⳵ y 21
冊册
共37兲
2
兩 a (1)
1 兩 ,
where
␣ 1 ⫽⫺
␣ 2⫽
6⫹b 2 关 ␻ s2 ⫺3 ␬ 共 3 ␬ ⫹1 兲兴
3 共 1⫹ ␬ b 2 兲
␻ s2 ⫺3 ␬ 共 3 ␬ ⫹1 兲
3 共 1⫹ ␬ b 2 兲
FIG. 3. Map of parameter plane, showing where the DSI equations govern
the sound–ultrasound interaction in bubbly fluids.
,
共38兲
.
From Eq. 共3兲 it follows that
a (1)
1 ⫽
p (1)
1
共 ␻ s2 ⫺3 ␬ 兲
a (0)
1 ⫽⫺
⫹O 共 ␧ 兲 ,
冋
册
1
3 ␣ 2 共 1⫹ ␬ b 2 兲 (1) 2
p (0)
兩 p 1 兩 ⫹O 共 ␧ 兲 .
1 ⫹
3␬
共 ␻ s2 ⫺3 ␬ 兲 2
共39兲
Substitution of Eqs. 共39兲 into Eq. 共38兲 leads to the cancella2
tion of the second derivatives of 兩 p (1)
1 兩 with respect to x 1
and y 1 . We then have the following zeroth-harmonic equation for p:
⳵ 2 p (0)
1
⳵ t 21
⫺c 2e
冉
⳵2
⳵ x 21
⫹
⳵2
⳵ y 21
冊
p (0)
1 ⫽
2
␣ ⳵ 2 兩 p (1)
1 兩
c 2g
⳵ t 21
,
共40兲
which is reduced to Eq. 共35a兲.
A. Dromions in bubbly fluids
When ␦ ⫽0 and ⌫⫽0, Eqs. 共35兲 can be transformed into
the canonical form.20,23 For this purpose, we perform the
substitutions L( ␰ , ␨ )→ ␦ ⫺1 ⳵ ⌿/ ⳵␰ and S̃⫽S exp兵i␦L0␶其 and
2
2
introduce the notation ␴ ⫽c ⫺2
e (c e ⫺c g ):
␴
⳵ 2⌿
⳵␰ 2
⫹
⳵ 2⌿
⳵␨ 2
⫽⫺
␣ ␦ ⳵ 兩 S̃ 兩 2
,
c 2e ⳵␰
⳵ S̃
⳵ 2 S̃
⳵ 2 S̃
⳵⌿
.
i ⫹ ␤ 2 ⫹% 2 ⫹ ␥ 兩 S̃ 兩 2 S̃⫽S̃
⳵␶
⳵␰
⳵␰
⳵␨
共41兲
It is known24 that Eqs. 共41兲 are nonintegrable for most
values of the coefficients. The exception is the case when
共41兲 can be represented as
␦1
⳵ 2Q
⳵x2
⫹
⳵ 2Q
⳵y2
⫽⫺ ␦ 2
⳵兩A兩2
,
⳵x
⳵A
⳵Q
⳵ 2A ⳵ 2A
i
⫹ ␯ 2 ⫹ 2 ⫽⫺ ␯ 兩 A 兩 2 A⫹A
,
⳵t
⳵x
⳵x
⳵y
with ␯ ⫽⫾1, ␦ 1 ⫽⫺ ␯ , ␦ 2 ⫽2.
共42兲
Depending on the sign of ␯ , Eqs. 共42兲 are divided into
the Davey–Stewartson I 共DSI兲 equations, of hyperbolicelliptic type ( ␯ ⫽1), and the Davey–Stewartson II 共DSII兲
equations, of elliptic-hyperbolic type ( ␯ ⫽⫺1). The integrability of the DSI and DSII equations by inverse scattering
transform was proved in numerous publications 共the reader is
referred to Ref. 35 for a list of relevant articles兲.
Since the coefficients ␤ and % in 共35兲 are always positive 关cf. 共21兲 and 共36兲兴, the DSI case is only possible for the
interacting waves in bubbly fluids. To show this, it is enough
to perform the substitutions Q⫽⌿, A⫽ ␥ 1/2S̃, x⫽ ␰ / ␤ 1/2, y
⫽ ␨ /% 1/2, t⫽ ␶ in Eqs. 共41兲. The DSI equations occur when
%
␦ 1 ⫽ 关 1⫺c 2g /c 2e 兴 ⫽⫺1,
␤
␦ 2⫽
␣␦%
␥␤ 1/2c 2e
⫽2.
共43兲
You might see before that the coefficients ␣ and ␦ were of
the same sign 共Fig. 2兲. Hence, to satisfy the first condition of
共43兲, ␥ should be positive. Then the radicand in the second
condition is also positive.
The intersection of the curves ␦ 1 ⫽⫺1 and ␦ 2 ⫽2 on the
parameter plane (k s , b), plotted in Fig. 3共a兲, proves the satisfaction of conditions 共43兲. Figure 3共b兲 shows two points of
this intersection (D 1 and D 2 ) that correspond to the volume
gas contents ␣ g01⬇2.04⫻10⫺4 , ␣ g02⬇1.42⫻10⫺4 and the
ultrasound frequencies ⍀ s1 ⬇0.11 MHz, ⍀ s2 ⬇0.14 MHz 共in
the case of a mixture of water with air bubbles of the radius
0.1 mm under normal conditions兲.
Among all solutions of the DSI equation, so-called ‘‘dromion’’ solution46 stands out. This solution represents a localized and exponentially decaying ultrasound wave A which
can scatter energy during the interaction with perturbations,
in contrast to usual soliton solutions. The dromion arises
when nonzero boundary conditions for the sound profile Q
are given and therefore exists due to energy exchange between sound and ultrasound waves.
Of even greater importance is the fact that the dromion
共ultrasound wave兲 travels along the trajectory determined by
the time dependence of the boundary conditions for the
sound wave 共at infinity兲. Setting the law that governs the
variation of these boundary conditions, one can control the
motion of the dromion.
Fokas and Santini46 showed that the DSI equations govern the evolution of N dromions (N is natural兲 for arbitrary
time-dependent boundary conditions for Q and any initial
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3590
Phys. Fluids, Vol. 13, No. 12, December 2001
D. B. Khismatullin and I. Sh. Akhatov
conditions for A decaying at infinity. To find a one-dromion
solution it is necessary to rotate the system of coordinates
through 45°: x→x̃⫹ỹ, y→x̃⫺ỹ. By introducing the variables U⬅( ⳵ Q/ ⳵ x⫺ 兩 A 兩 2 )/2 and V⬅( ⳵ Q/ ⳵ y⫺ 兩 A 兩 2 )/2 共the
tilde is omitted兲, Eqs. 共42兲 with ␯ ⫽1 can then be written in
the form
⳵ A ⳵ 2A ⳵ 2A
i ⫹ 2 ⫹ 2 ⫹ 共 U⫹V 兲 A⫽0,
⳵t
⳵x
⳵y
⳵U ⳵兩A兩2
⫽
,
⳵y
⳵x
A⫽
U 兩 y→⫺⬁ ⫽
V 兩 x→⫺⬁ ⫽
共44兲
⳵V ⳵兩A兩2
⫽
.
⳵x
⳵y
Provided that the boundary conditions for U on y and V on x
at ⫺⬁ are of the form of one-dimensional KdV solitons
8k r2 exp共 ␩ 1 ⫹ ␩ 1* 兲
关 1⫹exp共 ␩ 1 ⫹ ␩ 1* 兲兴 2
8l r2 exp共 ␩ 2 ⫹ ␩ *
2兲
2
关 1⫹exp共 ␩ 2 ⫹ ␩ *
2 兲兴
,
共45兲
,
Eqs. 共44兲 are easily integrated and give a one-dromion solution of the form47
␹ exp共 ␩ 1 ⫹ ␩ 2 兲
1⫹exp共 ␩ 1 ⫹ ␩ 1* 兲 ⫹exp共 ␩ 2 ⫹ ␩ 2* 兲 ⫹␭ exp共 ␩ 1 ⫹ ␩ 1* ⫹ ␩ 2 ⫹ ␩ 2* 兲
共46兲
.
B. Singular focusing
Here
␩ 1 ⫽ 共 k r ⫹ik i 兲 x⫹ 共 ⫺2k r k i ⫹i⍀ i 兲 t,
共47a兲
␩ 2 ⫽ 共 l r ⫹il i 兲 y⫹ 共 ⫺2l r l i ⫹i ␻ i 兲 t,
共47b兲
⍀ i ⫹ ␻ i ⫽k r2 ⫹k 2i ⫹l r2 ⫹l 2i ,
共47c兲
␹ ⫽2 共 2k r l r 共 ␭⫺1 兲兲 ,
共47d兲
1/2
and ␭, k r , k i , l r , l i are arbitrary real numbers 共parameters
of the solution兲. The spatial distributions of the ultrasound
envelope 兩 S̃ 兩 ⫽ ␥ ⫺1/2兩 A 兩 and of the sound profile L⫽(U⫹V
⫹2 兩 A 兩 2 )/(2 ␦ ), which correspond to the one-dromion solution given by Eqs. 共45兲–共47兲, are illustrated in Fig. 4.
The numerical experiments47 confirm the fact that the
motion of the dromion structure occurs along the trajectory
determined by the boundary conditions for the long wave.
For example, when the variables in Eqs. 共45兲 are changed as
␩ 1 ⫹ ␩ *1 ⫽k r x⫹⍀ r sin共 wt 兲 ,
␩ 2 ⫹ ␩ *2 ⫽l r x⫹ ␻ r cos共 wt 兲 ,
i.e., the crosspoint of the sound wave 共point C in Fig. 4兲
revolves around the origin, the dromion propagates quasistably around a circle.
If the coefficients ␤ , %, ␴ , ␣ ␦ are positive and
␥ ⬎⫺ ␣ ␦ /c 2e , the solution to Eqs. 共41兲 with the boundary
condition
兩 S̃ 兩 →0
for ␰ 2 ⫹ ␨ 2 →⬁,
tends to infinity for reasonably large amplitudes within a
finite time interval.25 This phenomenon is often referred to as
singular focusing or blowup instability of two-dimensional
wave perturbations. There are experimental data48 which
confirm the existence of the singular focusing in nonlinear
optics.
The singular focusing of the solutions to Eqs. 共41兲 is
corroborated by the analysis of conservation laws. From Eqs.
共41兲 it follows that the integrals:
M⫽
冕冕
⫹⬁
⫺⬁
兩 S̃ 兩 2 d ␰ d ␨ ,
共49兲
冉
冊
冊
冕冕 冉
冕冕 冋 冏 冏 冏 冏
冏 冏 冏 冏 冊册
冉
P x⫽
冕冕
P y⫽
⫹⬁
⫺⬁
⫹⬁
⫺⬁
⫹⬁
E⫽
⫺⬁
⫺
FIG. 4. One-dromion solution of the DSI equations with ␭⫽3, k r ⫽l r
⫽4/5, k i ⫽l i ⫽1/5.
共48兲
S̃ *
⳵ S̃
⳵ S̃ *
⫺S̃
d␰d␨,
⳵␰
⳵␰
共50兲
⳵ S̃
⳵ S̃ *
S̃ * ⫺S̃
d␰d␨,
⳵␨
⳵␨
2
␤
⳵ S̃
⳵ S̃
⫹%
⳵␰
⳵␨
2
␴ c 2e ⳵ ⌿ 2 c 2e ⳵ ⌿
1
␥ 兩 S̃ 兩 4 ⫹
⫹
2
␣ ␦ ⳵␰
␣ ␦ ⳵␨
2
d␰d␨,
共51兲
are conserved. The first integral is the ‘‘mass’’ of the short
wave. The next two integrals are the ␰ - and ␨ -components of
the short-wave ‘‘momentum.’’ We obtain 共49兲 and 共50兲 by
multiplying the second equation of 共41兲, respectively into
2S̃ * , S̃ ␰* , and S̃ ␨* , then taking the real parts of the results,
and integrating them with respect to ␰ and ␨ between ⫺⬁
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Phys. Fluids, Vol. 13, No. 12, December 2001
Sound–ultrasound interaction in bubbly fluids
3591
FIG. 7. Nonlinear distortions of the ultrasound wave packet. The values of
parameters are listed in Fig. 6. The curves labeled 1, 2, 3, and 4 are for the
instants of time t⫽0.4557, 0.4652, 0.4747, and 0.4842 s, respectively.
FIG. 5. Regions of the blowup instability ( ␥ f ⫽ ␥ ⫹ ␣ ␦ /c e2 ).
and ⬁ using the boundary condition 共48兲. The integral 共51兲
may be interpreted as the ‘‘total energy’’ of interacting long
and short waves. The ‘‘energy’’ can be negative under the
condition of singular focusing and large initial amplitudes of
ultrasound waves. The steps of deriving 共51兲 are following:
共i兲 The second equation of 共41兲 is multiplied into 2S̃ ␶* and
the real part of the result is considered; 共ii兲 the real part
contains the interaction term
⳵ 兩 S̃ 兩 2 ⳵ ⌿
,
⳵ ␶ ⳵␰
which is calculated using the first equation of 共41兲; 共iii兲 the
obtained expression is integrated with the use of 共48兲.
From 共41兲 it follows also the relation:
d 2I
dt 2
⫽8E,
冕冕 冉
⫹⬁
I⫽
⫺⬁
冊
␰2 ␨2
⫹
兩 S̃ 兩 2 d ␰ d ␨ ,
%
␤
共52兲
where I is the ‘‘inertia moment’’ 共a positive-definite magnitude兲. The derivation of 共52兲 is not trivial and is shown in
detail by Ghidaglia and Saut24 共see also Ref. 25兲. When E
⬍0, the ‘‘virial theorem’’ 共52兲 causes the ‘‘inertia moment’’
to vanish in a finite time. Physically, this means that the
ultrasonic perturbation distributed initially in space will be
concentrated at some spatial point with time. Since the mass
of the ultrasound wave is conserved, this process leads to
increasing the wave amplitude at the spatial point and, finally, results in the blowup instability.
In the bubbly fluid, the coefficients ␤ , % are always
positive, and ␣ and ␦ are of the same sign. Hence, the singular focusing of ultrasound is possible in the bubbly mixtures if49
c e ⬎c g ,
␥ ⬎⫺ ␣ ␦ /c 2e .
共53兲
The regions of the blowup instability, determined by condition 共53兲, are plotted in Fig. 5.
VIII. NUMERICAL ANALYSIS
A. Degenerate resonant interaction
The models 共28兲 and 共32兲 are integrated numerically using a three-layer explicit scheme with a fourth order of approximation with respect to the spatial coordinate proposed
earlier50 for the numerical solution of the KdV equation 共see
details in the Appendix兲. Periodic boundary conditions and
an initial condition of the form
L 兩 ␶ ⫽0 ⫽L 0 共 1⫹⌬L 关 1⫺cos ␰ 兴 兲 ,
S r 兩 ␶ ⫽0 ⫽S 0 共 1⫹⌬S 关 1⫺cos ␰ 兴 兲 ,
S i 兩 ␶ ⫽0 ⫽0,
共54兲
where S r ⫽Re(S); S i ⫽Im(S), are used in the integration.
The results of numerical investigation of 共28兲 are represented in Figs. 6 – 8. Figure 6 shows the spatial distributions
of the ultrasound envelope and the sound profile 共damping is
not considered兲. It is clear that a sinusoidal profile for 兩 S 兩 is
FIG. 6. Nondissipative evolution of 共a兲 the sound wave, p L ⫽␧ l p 0 L, and 共b兲
the ultrasound wave packet, 兩 p S 兩 ⫽␧ s p 0 兩 S 兩 , according to Eqs. 共24兲. Water
with air bubbles of the radius 1 mm under normal conditions and ␧⫽0.1,
L 0 ⫽S 0 ⫽5, ⌬L⫽0.25, ⌬S⫽0.05, ␣ g0 ⫽3⫻10⫺4 , ␮ ⫽0. The curves labeled
1, 2, 3, 4, 5, 6, and 7 are for the instants of time t⫽0.019, 0.171, 0.209,
0.294, 0.437, 0.532, and 0.722 s, respectively. Here t⫽ ␶ /␧ 2 and x
⫽(a 0 ␣ ⫺1/2
g0 /␧) ␰ .
FIG. 8. The maximal value of the ultrasound amplitude versus time. The
other parameters are the same as in Fig. 6.
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3592
Phys. Fluids, Vol. 13, No. 12, December 2001
FIG. 9. Tendency to a shock wave formation in model 共26兲. Here ␧⫽0.1,
L 0 ⫽S 0 ⫽5, ⌬L⫽⌬S⫽0.05, p 0 ⫽0.1 MPa, ␬ ⫽1.4. The curves labeled 1, 2,
and 3 are for the instants of time t⫽0, 0.133, and 0.285 s.
formed. The amplitude of this profile increases up to the
instant of time t⫽0.294 s and then sharply decreases at t
⫽0.437 s, and, after a certain time, it increases again, reaching a maximum value at t⫽0.722 s 关Fig. 6共a兲兴. In the case of
sinusoidal initial conditions ultrasound has no action on
sound and the sound wave propagates in accordance with the
KdV equation 共19兲. It is important to note that an increase in
the ultrasound amplitude occurs during the formation of a
second hump in the sound profile 共the two-soliton solution of
the KdV equation兲 and a decrease happens when the hump
disappears 关Fig. 6共b兲兴. From this it follows that the formation
FIG. 10. Evolution of the sound perturbation p L ⫽␧ l p 0 L. The parameters
are given in the legend and in Fig. 9.
D. B. Khismatullin and I. Sh. Akhatov
FIG. 11. Spatial distribution of the sound perturbation according to Eqs.
共26兲 for different values of the damping coefficient.
of a two-humped profile is accompanied by a significant return flow of energy from sound to ultrasound, on account of
which, in spite of strong dispersion, the amplitude of the
ultrasound wave increases. The return flow of energy is provided by the interaction terms L ⳵ S/ ⳵␰ and S ⳵ L/ ⳵␰ . When L
increases or, at least, does not decrease at some spatial point
␰ ⫽ ␰ , these terms ensure a positiveness of the derivative
*
⳵ S/ ⳵ ␶ and, hence, increasing the ultrasound amplitude at the
point ␰ in time. Going through the periods of growth and
*
fall, this amplitude grows on the whole which is seen from a
comparison of the ultrasonic distribution at the instants of
time t⫽0.209 s and t⫽0.722 s 关curves 3 and 7 in Fig. 6共a兲兴.
The essential smallness and briefness of the nonlinear distortions in the ultrasound profile should be noted. These distortions appear when the amplitude of ultrasound reduces 关cf.
Fig. 7兴. Consequently, we have the linear intensification of
ultrasound by sound here.
The results obtained can be given the following physical
interpretation. Suppose that quasimonochromatic highfrequency 共ultrasonic兲 and low-frequency 共sonic兲 oscillations
are excited in a bubbly fluid. In general, these perturbations
will propagate in the medium in the form of an ultrasound
wave packet and a sound wave. When the long-wave–shortwave resonance condition is satisfied, interaction occurs
which obeys the Zakharov equations 共24兲. This system describes the formation of a soliton structure for the given
boundary-initial conditions 共in the nondissipative case兲.45
Obviously, the soliton is a consequence of significant nonlinear distortions in the ultrasound profile. However, an isolated
ultrasound frequency exists in a bubbly fluid, that is, the
frequency of ‘‘degeneracy’’ at which the resonant interaction
has a completely different character. First, the sound wave
propagates independently of the ultrasound wave and obeys
the KdV equation. Second, due to the action of the sound
wave on the ultrasound wave a sinusoidal profile of the ultrasound envelope is formed. This profile experiences negligible and short-term nonlinear distortions. This means that
practically all the energy of ultrasound will be contained in
its first harmonic. Third, due to the flow of energy from the
sound wave to the ultrasound wave, the growth of the shortwave amplitude takes place.
What happens if we add damping? Restrict our attention
to very small damping ( ␮ * ⫽␧ 3 ␮ ) because small one ( ␮ *
⫽␧ 2 ␮ ) suppresses any interaction in this model 共cf. above兲.
Of course, the very weak damping leads to the attenuation of
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Phys. Fluids, Vol. 13, No. 12, December 2001
Sound–ultrasound interaction in bubbly fluids
3593
FIG. 12. Nonresonant defocusing of ultrasound and
sound in a bubbly fluid (S 0 ⫽25, ␬ ⫽1.4).
ultrasonic perturbations. If ␮ is large enough, the amplitude
of the short wave decreases exponentially in time 共thin solid
line in Fig. 8兲. The interaction between sound and ultrasound
waves appears for a smaller damping coefficient. Due to the
interaction, the attenuation curve of the ultrasound amplitude
becomes of graduated type 共thick solid line in Fig. 8兲. A
slower attenuation is due to increasing the ultrasound amplitude 共in the nondissipative case兲.
The absolutely different situation is observed for model
共32兲. Figure 9 shows the sonic distribution with respect to x
at ␮ ⫽0 at different instants of time. It is clear that sound
propagates with its profile becoming steeper since, when no
account is taken of terms of order ␧, the equation for the
sound wave is the Hopf equation 共34兲.
Figures 10 and 11 represent the subsequent evolution of
the sonic and ultrasonic perturbations in the presence 共thick
line兲 and absence 共thin and dashed lines兲 of weak damping. A
shock profile is not formed. This is due to the contribution of
terms of order ␧ in the first equation of 共32兲. If the ultrasonic
perturbations are not generated (S 0 ⫽0), the sound wave will
propagate in accordance with the strongly nonlinear KdV
equation
⳵L
⳵L3
⳵L2
⳵ 3L
⫹␴
⫹␧␵
⫹␧ ␹ 3 ⫽0
⳵␶
⳵␰
⳵␰
⳵␰
共55兲
共Fig. 10, dashed line兲. But if S 0 ⬎0, the sound profile fails
共Fig. 10, thin line兲. Thus, the nondissipative interaction between high intensity sound and ultrasound results in nonlinear instability in the case of resonant degeneracy. Note that
this instability appears due to the tendency to a shock wave
formation and the presence of the terms with cubic nonlinearity, % ⳵ (L 兩 S 兩 2 )/ ⳵␰ and ˜␦ L 2 S, in Eqs. 共32兲. The values of
the sound profile are quite different in two close points on
the shock. This difference is doubled and transmitted to the
ultrasound wave due to the term ˜␦ L 2 S and leads to the further steepness of the ultrasound profile. Due to the term
% ⳵ (L 兩 S 兩 2 )/ ⳵␰ the originating jump in S influences the sound
wave and, finally, results in the nonlinear instability.
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3594
Phys. Fluids, Vol. 13, No. 12, December 2001
D. B. Khismatullin and I. Sh. Akhatov
FIG. 13. Defocusing of ultrasound in the resonant case: 共a兲 the ultrasound
envelope 兩 S 兩 versus ␰ ( ␨ ⫽0) at different instants of time; 共b兲 comparison of
the nonresonant 共thick curve兲 and resonant 共dashed curve兲 ultrasound profiles at ␶ ⫽1.
Weak damping suppresses the nonlinear instability and
results in insignificant decreasing the amplitude and velocity
of a solitary sound wave. The disappearance of the instability
comes from the fast attenuation of the ultrasonic signal: The
contribution of S into L becomes small even during the
steepness of the ultrasonic distribution. Nevertheless, if we
reduce the damping coefficient, the nonlinear instability will
arise again 共Fig. 11, thick curve兲. This is because the ultrasound amplitude decreases slowly now, and nonlinear distortions are large enough for the destruction of the sound profile.
damping suppresses this instability until ␮ exceeds some
cutoff value. It should be noted that a similar cut-off influence of damping on singular focusing was observed for the
generalized KdV equation with nonlinearities of fourth
order.53
The singular focusing of nondissipative interaction is
evident in the fourth 共unstable resonant兲 case. Now the focusing is accelerated and accompanied by the strong deformation of the ultrasound profile 共Fig. 15兲. Also, the amplitude threshold for singular focusing reduces to S 0 ⫽20.
Hence the long-wave–short-wave resonance enhances the
probability of singular focusing of ultrasound in bubbly fluids.
IX. POSSIBLE APPLICATIONS
In conclusion, we would like to outline possible applications of such sound–ultrasound interactions.
First, the phenomenon of degeneracy of the interaction
can be used for ultrasound diagnostics of weakly viscous
bubbly fluids in narrow channels. Narrow channels are essential here in order that transverse instabilities do not
evolve. If we find some value of the degeneracy frequency
experimentally, we can evaluate the radius of the bubble,
a 0d , and the volume gas content ␣ g0d . Indeed, from condition 共29兲 and formulas 共2兲 and 共4a兲 it follows that ␣ g0d depends only on the ambient pressure p 0 , mass density of a
carrier liquid, ␳ l0 , speed of sound in the liquid, C l , and
polytropic exponent ␬ :
B. Singular focusing
The results of numerical analysis of Eqs. 共35兲 are illustrated in Figs. 12–15. The first equation is solved by the
Fourier transform method,51 the second equation by the
alternating-direction implicit method.52 The boundary and
initial conditions are as follows:
L共 ␰,␨ 兲,
S 共 ␶ , ␰ , ␨ 兲 ⫽0,
for ␰ 2 ⫹ ␨ 2 →⬁
共56a兲
and
S 共 ␶ ⫽0,␰ , ␨ 兲 ⫽S 0 exp兵 ⫺ 共 ␰ 2 ⫹ ␨ 2 兲 其 .
共56b兲
Here S 0 takes real values. The coefficients in 共35兲 are selected so that c e ⬎c g 关then the first equation of 共35兲 is of
elliptic type兴. We analyze four cases: 共1兲 k s ⫽0.5, b⫽0.4; 共2兲
k s ⫽1.036, b⫽0.4; 共3兲 k s ⫽0.5, b⫽0.25; 共4兲 k s ⫽0.574, b
⫽0.25. The first two cases are considered on the assumption
that ␮ ⫽0. The first case corresponds to the stable nonresonant interaction because c e ⫽c g and point 共0.5; 0.4兲 lies in
the stable region 共cf. Fig. 5兲.
Numerical analysis confirms analytical results. The
stable interaction leads to defocusing of sound and ultrasound waves 共Fig. 12兲. In the second case we are also in the
stable region but near the resonant curve (c g →c e ). The numerical solution of 共35兲 defocuses again 关Fig. 13共a兲兴. But
now the defocusing slows down and is accompanied by the
modification of the wave profile 关Fig. 13共b兲兴.
In the third case the unstable nonresonant interaction
takes place. Indeed, with no damping the high intensity ultrasound wave (S 0 ⬎40) is shrinking and growing in time
共Fig. 14兲. This results in the blowup instability later. Weak
␣ g0d ⫽
p 0 ␬ 2 共 3 ␬ ⫹2 兲 2
␳ l0 C 2l 关 3 ␬ 共 ␬ ⫹1 兲 ⫹1 兴
.
If ␣ g0d is known, there is no problem to find a 0d because
a 0d ⫽
3
2 ␲ f sd
冑
p 0 ␬ 共 ␬ ⫹1 兲
,
␳ l0 共 1⫺ ␣ g0d 兲
where f sd is the degeneracy frequency. Of course, actual
bubbly fluids are polydispersed. This requires to include the
bubble size distribution in the equations of motion. There are
two changes in the equations for perturbations 共3兲 if the bubbly fluid becomes polydisperse 共see Ref. 16兲. First, we have
N
␣
兺 j0 共 1⫹a j 兲 3 ,
j⫽2 ␣ g0
instead of (1⫹a) 3 in Eq. 共3b兲. Here N⫺1 is the number of
disperse 共or bubble兲 fractions (N⫽1 corresponds to the dispersive medium兲, ␣ j0 and a j are the initial volume content
and perturbation in the bubble radius for jth fraction, ␣ g0
⫽ 兺 Nj⫽2 ␣ j0 is the total initial volume gas content. Second, in
Eq. 共3c兲, a is replaced by a j and the first two terms are
multiplied by
␰ j⫽
a j0
a
*
(a j0 is the initial bubble radius for jth fraction, a is a cer*
tain representative radius of bubbles兲. There are N⫹1 equations for perturbations at all: The first is for the pressure
perturbation, the second is for the density perturbation, and
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Phys. Fluids, Vol. 13, No. 12, December 2001
Sound–ultrasound interaction in bubbly fluids
3595
FIG. 14. Nonresonant singular focusing of ultrasound
and sound in a bubbly fluid (S 0 ⫽30; ␬ ⫽1.4; ␮ ⫽0).
the remaining N⫺1 equations are for the perturbations in
bubble radii. Anyway, the equations have the same form, and
hence the above results should be valid for polydisperse bubbly fluids. For example, there exists the long-wave–short-
FIG. 15. Spatial distribution of the ultrasound envelope 兩 S 兩 at different
instants of time in the case of resonant singular focusing. Here S 0 ⫽20; ␬
⫽1.4, ␮ ⫽0.
wave resonance and the nonresonant one-dimensional dynamics is governed by the nonlinear Schrödinger equation.16
The main problem is to ‘‘catch’’ the degeneracy frequency. To do this the following ideas can be exploited. Usually, if we create ultrasound and sound in bubbly fluids simultaneously, so that resonance condition 共10兲 is satisfied,
these waves will interact according to the Zakharov equations with damping 共24兲. The result of this interaction is the
formation and the subsequent decay of the ultrasound envelope and the appearance of an additional sonic 共long-wave兲
component proportional to a square of the ultrasound envelope modulus. Since weak damping leads to decreasing the
ultrasound amplitude, the additional long-wave component
will be also damped out but not so fast as in the degenerate
case. Under degeneracy condition 共29兲, ultrasound will not
influence the long wave essentially. This gives us a chance to
verify the degeneracy phenomenon experimentally. The experimental setup can be designed on a basis of the
difference-frequency sound generation technique.12,54
Second, the dromion property to travel on the tracks,
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3596
Phys. Fluids, Vol. 13, No. 12, December 2001
defined by the evolution of sound wave, can be drawn on to
control ultrasonic action processes in bubbly fluids. If the
ultrasonic disturbance is created in a liquid with gas bubbles,
the parameters of which satisfy the existence of the dromion,
there is then a possibility to focus the disturbance at some
point of space and ‘‘remotely’’ control its motion by sound
共on significant distance from the focus point兲.
Third, singular focusing of ultrasound is possibly a very
dangerous phenomenon, especially in the case of longwave–short-wave resonance. It may result in explosions,
violent cavitation in a liquid with microbubbles, for example,
in human tissue, blood, and ocean water. Currently, the wide
range of potential medical applications of high intensity focused ultrasound 共HIFU兲, such as lithotripsy,55 cancer
therapy,56 and gene therapy,57 is being developed. It should
be noted that HIFU must be exploited in tissue in circumstances where the blowup instability condition is not satisfied.
The above-listed applications of the results invite further
theoretical and experimental investigations. Our following
efforts will be aimed at the elucidation of more complex
theoretical models and the performance of required experiments.
ACKNOWLEDGMENTS
APPENDIX: NUMERICAL METHOD FOR SOLVING
DEGENERATE EQUATIONS
The equations for resonant degenerate interaction 共28兲
and 共32兲 have been solved numerically using a three-layer
explicit scheme with a fourth order of approximation with
respect to the spatial coordinate.50 In the appendix, the finitedifference approximation of only Eqs. 共28兲 is presented. The
discrete equations of the second model are too cumbersome
to be given here. According to the scheme, the equation for
the long-wave profile L looks as follows:
⫺
␴ ⌬t n n
L 关 L ⫺8L nj⫹1 ⫹8L nj⫺1 ⫺L nj⫺2 兴
3⌬x j j⫹2
␭⌬t
3 共 ⌬x 兲
Si nj 关 Sr nj⫹2 ⫺16Sr nj⫹1 ⫹30Sr nj
2
⫺16Sr nj⫺1 ⫹Sr nj⫺2 兴 ⫹
␭⌬t
3 共 ⌬x 兲
Sr nj 关 Si nj⫹2
2
⫺16Si nj⫹1 ⫹30Si nj ⫺16Si nj⫺1 ⫹Si nj⫺2 兴
⫹
␹ ⌬t
4 共 ⌬x 兲 3
⫹8L nj⫺2 ⫺L nj⫺3 兴 .
关 L nj⫹3 ⫺8L nj⫹2 ⫹13L nj⫹1 ⫺13L nj⫺1
共A1兲
Because the short-wave envelope is complex-valued (S
⫽Sr⫹iSi), Eq. 共28b兲 breaks down into the following discrete equations:
␮ ⌬t n n
L 关 Sr j⫹2 ⫺8Sr nj⫹1 ⫹8Sr nj⫺1
6⌬x j
⫽Sr n⫺1
⫺
Sr n⫹1
j
j
⫺Sr nj⫺2 兴 ⫺
␯ ⌬t n n
Sr 关 L ⫺8L nj⫹1 ⫹8L nj⫺1
6⌬x j j⫹2
⫺L nj⫺2 兴 ⫹2 ␥ ⌬tSi nj 关共 Sr nj 兲 2 ⫹ 共 Si nj 兲 2 兴
⫹
˜␤ ⌬t
6 共 ⌬x 兲
2
关 Si nj⫹2 ⫺16Si nj⫹1 ⫹30Si nj ⫺16Si nj⫺1
⫹Si nj⫺2 兴 ⫹
␩ ⌬t
4 共 ⌬x 兲 3
关 Sr nj⫹3 ⫺8Sr nj⫹2 ⫹13Sr nj⫹1
⫺13Sr nj⫺1 ⫹8Sr nj⫺2 ⫺Sr nj⫺3 兴
共A2兲
and
␮ ⌬t n n
L 关 Si j⫹2 ⫺8Si nj⫹1 ⫹8Si nj⫺1
6⌬x j
⫽Si n⫺1
⫺
Si n⫹1
j
j
⫺Si nj⫺2 兴 ⫺
This material is based upon work supported by the European Commission under INCO-Copernicus program No.
ERBIC15CT980141 and the North Atlantic Treaty Organization under Grant No. 0000779. Any opinions, findings, conclusions or recommendations expressed in this publication
are those of the authors and do not necessarily reflect the
view of NATO.
⫽L n⫺1
⫹
L n⫹1
j
j
D. B. Khismatullin and I. Sh. Akhatov
␯ ⌬t n n
Si 关 L ⫺8L nj⫹1 ⫹8L nj⫺1
6⌬x j j⫹2
⫺L nj⫺2 兴 ⫺2 ␥ ⌬tSr nj 关共 Sr nj 兲 2 ⫹ 共 Si nj 兲 2 兴
⫺
˜␤ ⌬t
6 共 ⌬x 兲 2
关 Sr nj⫹2 ⫺16Sr nj⫹1 ⫹30Sr nj ⫺16Sr nj⫺1
⫹Sr nj⫺2 兴 ⫹
␩ ⌬t
4 共 ⌬x 兲 3
关 Si nj⫹3 ⫺8Si nj⫹2 ⫹13Si nj⫹1
⫺13Si nj⫺1 ⫹8Si nj⫺2 ⫺Si nj⫺3 兴 .
共A3兲
Here subscripts and superscripts denote spatial and temporal
locations, ⌬t and ⌬x are the time and spatial steps, t⫽t 3
⫽␧ ␶ , x⫽ ␰ , L nj ⫽L(x j ,t n ),Sr nj ⫽Sr(x j ,t n ), Si nj ⫽Si(x j ,t n ),
x j ⫽( j⫺1)⌬x, t n ⫽n⌬t, and ˜␤ ⫽ ␤ /␧. The indexes j and n
take the values from 1 and 0 to J and N, respectively.
The boundary conditions are periodic
L n⫹1
⫽L n⫹1
,
0
J
n⫹1
L J⫹1
⫽L n⫹1
,
1
n⫹1
L J⫹2
⫽L n⫹1
,
2
n⫹1
⫽L n⫹1
,
L J⫹3
3
⫽Sr n⫹1
,
Sr n⫹1
0
J
n⫹1
Sr J⫹1
⫽Sr n⫹1
,
1
n⫹1
⫽Sr n⫹1
,
Sr J⫹2
2
n⫹1
Sr J⫹3
⫽Sr n⫹1
,
3
⫽Si n⫹1
,
Si n⫹1
0
J
n⫹1
Si J⫹1
⫽Si n⫹1
,
1
n⫹1
⫽Si n⫹1
,
Si J⫹2
2
n⫹1
Si J⫹3
⫽Si n⫹1
.
3
共A4兲
The stability analysis for the numerical scheme has been
performed using the Neumann method.52 According to this
method, we assume that
⫽L ⬘ e at⫹ikx ,
L n⫺1
j
Sr n⫺1
⫽Sr ⬘ e at⫹ikx ,
j
⫽Si ⬘ e at⫹ikx ,
Si n⫺1
j
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Phys. Fluids, Vol. 13, No. 12, December 2001
L nj ⫽L n⫺1
e a⌬t ,
j
Sr nj ⫽Sr n⫺1
e a⌬t ,
j
n⫺1 ik⌬x
e
,
L n⫺1
j⫹1 ⫽L j
Sound–ultrasound interaction in bubbly fluids
Si nj ⫽Si n⫺1
e a⌬t ,
j
n⫺1 ik⌬x
Sr n⫺1
e
,
j⫹1 ⫽Sr j
n⫺1 ik⌬x
e
.
Si n⫺1
j⫹1 ⫽Si j
Substitution of Eqs. 共A4兲 into the finite-difference equations
, Sr n⫺1
, and
leads to a system of algebraic equations in L n⫺1
j
j
n⫺1
for each of the models. From the compatibility condiSi j
tion we obtain a polynomial equation in exp(a⌬t). The stability criterion follows from solving this equation and is
based on the fact that the linear instability evolves when
exp(a⌬t)⬎1.
The stability analysis for Eqs. 共A1兲–共A3兲 results in the
polynomial equation of the fourth order which has rather
unwieldy solutions. We have examined the approximate solutions to this equation using Mathematica and obtained the
following stability criterion at ˜␤ ⬇4.9 and ␩ ⬇0.06:
冉
⌬t⭐min 0.042⌬x 3 ,
冊
0.216⌬x 3
.
␹
共A5兲
The stability criterion for the finite-difference equations of
共32兲 is even simpler
⌬t⭐min
1
冉
冊
3 冑2⌬x 2 0.216⌬x 3
,
.
32␤
␧␹
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